Asymptotic analysis of dipolar mixed modes in red giant stars

Recent observations of oscillations in subgiant and red giant stars have given a great impact on our understanding of the structure and evolution of the stars. The most important property of the oscillations is the detection of dipolar mixed modes, which can probe the stellar structure both in the core and in the envelope. In order to understand those modes from a theoretical point of view, an asymptotic analysis is performed under the assumption that the oscillations are composed of short-wavelength waves both in the core and in the envelope. Compared to the conventional analysis in the theory of stellar oscillation, the present analysis has the following new aspects : (i) the perturbation to the gravitational potential is fully taken into account (namely, the Cowling approximation is not assumed) ; (ii) the strong interaction between the oscillations in the central gravity-wave (G) cavity and those in the envelope acoustic-wave (P) cavity is considered. The formulae that are derived in the analysis include the quantisation condition, which determines the eigenfrequencies of the modes, and the amplitude ratio between the G cavity and the P cavity. Based on these, the origin and implication of the empirical phase shift, which is usually expressed as ’epsilon’, in the frequency formula is particularly discussed. The results should be useful in interpreting the observed oscillation spectra.